The magnetic field repels diamagnetic bodies. The force which acts on a diamagnetic body is proportional to its volume, magnetic susceptibility and the gradient of the magnetic field energy density (i.e. the square of the magnetic field). The last parameter (let's name it G) characterizes levitation conditions. To levitate water, one needs a threshold value of G = 1400 TT/m .

To increase the force density one can either increase the magnetic field (powerful superconducting magnets) or increase the field gradient (i.e. go to the micro/nanoscale). For example, in a 15 Tesla magnet with a typical internal bore size of 10 cm the parameter G is about 2000 TT/m. In our microlevitation device the field on the magnet surface is 0.5 Tesla. With the distance between magnets of the order of 100 microns, the same parameter G is of the order of 5000 TT/m. It seems feasible to decrease the size by at least two orders of magnitude with a corresponding increase of G. Another way to increase the force density is to use a paramagnetic environment. We have increased the force density by additional 3 orders of magnitude in a paramagnetic solution [2].

At the nanoscale it is in many cases very important to prevent direct contact between nanoparticle(s) and macroscopic bodies. When a particle "sticks" to the surface of another particle or the the surrounding walls it is usually very difficult to remove it. Can we achieve this contact-less manipulation? All forces we have at hand (gravitational and/or electromagnetic) have one thing in common - they are Coulomb-like. Thus we need to build stable traps with Coulomb forces. Is it possible?

Surprisingly this problem was first discussed already in 1842 by Earnshaw [3] in connection with an even more exciting problem: can one build matter with particles interacting with Coulomb forces? The answer was a definite NO. The Earnshaw theorem was the remarkable result and have posed a problem which was solved almost a century later by quantum mechanics. In 1872 Lord Kelvin [4] recognized that it is possible to get stable equilibrium in the case of diamagnetic bodies (diamagnetism is a quantum phenomena). Indeed, a diamagnetic body is repelled by a magnetic field, and the force is proportional to the gradient of the magnetic field energy. As a result a diamagnetic body can be captured by a trap with a region of weak field (small magnetic field energy density) surrounded by a region of strong field (large magnetic field energy density). Indeed, the gradient is directed toward the trap center. Formally, a dielectric body in a dielectric media with a higher dielectric susceptibility should behave similar to a diamagnetic body. However, the omnipresent stray electric charges, even at very low concentration, strongly limit this approach. In contrast to an electric field, there appears to be no such thing as a single free magnetic charge. As a result manipulation tools based on diamagnetism can operate even in a conducting media, i.e. the media (water based solutions) in use in biochemistry or biology. Of course, diamagnetic levitation has no analogous electric competitor in air/vacuum. Note, that a charged droplet in capacitor (Millikan experiment) can not be in stable equilibrium due to Earnshaw theorem.

A possible solution for an electric field is to use a high frequency AC field, e.g. light, to create a non-contact manipulator. This approach has been realized with optical tweezers (see review [5]) which appeared almost thirty years ago and have been successfully used to manipulate micron scale objects like cells or beads inside a solution [5]. Though levitation in air of micron size droplets was demonstrated, it needs a rather large power. To manipulate nanoparticles, optical tweezer need even more power which results in the particle overheating, even melting. This limits application of optical tweezers with micron size particles. In contrast to tweezers the diamagnetic trap has zero power consumption.

How small can be particles captured by diamagnetic trap? Rough estimate one can get by comparing potential energy in magnetic field with thermal fluctuations. Such estimates, discussed in [2,6], gives for materials with susceptibility close to those of water (i.e. organics) the smallest volume of the order of 1000,000 nm3, for graphite this volume is 20 times smaller and is close to the large nanotube volume. This volume can be decreased by two orders of magnitude either by using a paramagnetic environment (solution) or by decreasing temperature to the liquid helium temperature. Thus it is feasible to manipulate nanotubes at helium temperature or at room temperature in a paramagnetic solution. Another intriguing possibility is to use paramagnetic liquid oxygen.

Coming back to the nice flying creature, it is worth noting that the magnetic field configuration used for frog levitation is actually not suitable for manipulation purposes. Gravity is essential in creating a potential well for the frog, but it is not an acceptable tool at the nanoscale. However, magnetic trap can not compete with flying frog in one thing: frog in mid-air is inspiring!

References

1. M. D. Simon and A. K. Geim, J. Appl. Phys. 87, 6200 (2000).

2. I. F. Lyuksyutov, A. Lyuksyutova, K.D.D. Rathnayaka and D.G. Naugle, Modern Phys. Lett. B 17, 935 (2003).

3. S. Earnshaw, Trans. Cambridge Philos. Soc. 7, 97 (1842).

4. W. Thomson, Reprint of Papers on Electrostatics and Magnetism, (Mac-

5. A. Ashkin, IEEE J. Sel. Top. Quantum Electron. 6, 841 (2000).

6. I. F. Lyuksyutov, Modern Phys. Lett. B 16, 569 (2002).